Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory

This work originated in the try to understand the results from B. Kostant's recent paper on a "cubic Dirac operator" from the point of view of differential geometry, and in particular the well-established Dirac operator techniques one has therein. At the same time, we became interested in invariant metric connections whose torsion (viewed as a (0,3)-tensor) is a 3-form as a means to construct solutions to the common sector of type II string theories. It turns out that the former is helpful for the latter, with the concept of connections with totally skew symmetric torsion on some special homogeneous spaces as the unifying principle.